Problem: Determine how many solutions exist for the system of equations. ${-x+y = -4}$ ${-3x+3y = -12}$
Answer: Convert both equations to slope-intercept form: ${-x+y = -4}$ $-x{+x} + y = -4{+x}$ $y = -4+x$ ${y = x-4}$ ${-3x+3y = -12}$ $-3x{+3x} + 3y = -12{+3x}$ $3y = -12+3x$ $y = -4+x$ ${y = x-4}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = x-4}$ ${y = x-4}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-x+y = -4}$ is also a solution of ${-3x+3y = -12}$, there are infinitely many solutions.